KiRSTiNE Smith 
75 
(3) When we introduce = 0, = yv^ and jx^ = yw* in (91) we get 
< = ^(l + 2ayr.2+a>'')^-J^,. 
This for constant is a minimum when y = 1 and then equal to 
a::. = ^(l + 2ai,2 + aV)^ (92). 
= ±- when possible, that is for a ^ 1 determines a minimum, while for 
a 
a < 1, a'i^ reaches its lowest value for = 1. From (92) we find for a >- I the 
minimum 
. 4a, 
N 
and for a < 1 the minimum 
both formulae giving o;,^ = ^ . 4 for a = 1. 
Our results are accordingly : 
when a > 1, o;,^ is a minimum and equal to . 4a for two equally big groups of 
observations at a; = ± - or for any distribution with the same and ^4 , 
and when a < 1, al^ is a minimum and equal to (1 + a)^ for two equally big 
groups of observations at x = ±1. 
We see that for a 5 0 two equally big groups of observations at ± 1 make both 
al^ and o-^^ minima and these groups in addition form the distribution for which 
has the lowest maximum within the possible range of observations. 
(4) For a. function of the second degree 
y = ao + ttjX + a^x^ , 
with the standard deviations of observations 
Sy = CT (1 + ax^), a > — 1, 
and therefore ^ ^ ^ ^"'^^ ^ 
we find^ from (89), 
" (1 + 2a/., + aV4) . . ^^t/^ (93). 
t^zfJ-i - /X, - fir, + ^/Xi/x.^/Xg - /Aj/i4 
and < = ^ (1 + 2aM, + a^f^,) . , f (95) 
